Trace maps on rigid Stein spaces

Abstract We provide a relative version of the trace map from the work of Beyer, which can be associated to any finite étale morphism $$ X \rightarrow Y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> of smooth rigid Stein spaces and which then relates the Serre duality on X with the Serre duality on Y . Furthermore, we consider the behaviour of any rigid Stein space under (completed) base change to any complete extension field and deduce a commutative diagram relating Serre duality over the base field with the Serre duality over the extension field.